On the Ill-posedness for the Navier–Stokes Equations in the Weakest Besov Spaces

It was proved in Iwabuchi and Ogawa (J Elliptic Parabol Equ 7(2):571–587, 2021) that the Cauchy problem for the full compressible Navier–Stokes equations of the ideal gas is ill-posed in \(\dot{B}_{p, q}^{2 / p}(\mathbb {R}^2) \times \dot{B}_{p, q}^{2 / p-1}(\mathbb {R}^2) \times \dot{B}_{p, q}^{2 / p-2}(\mathbb {R}^2) \) with \(1\le p\le \infty \) and \(1\le q<\infty \). In this paper, we aim to solve the end-point case left in [17] and prove that the Cauchy problem is ill-posed in \(\dot{B}_{p, \infty }^{d / p}(\mathbb {R}^d) \times \dot{B}_{p, \infty }^{d / p-1}(\mathbb {R}^d) \times \dot{B}_{p, \infty }^{d / p-2}(\mathbb {R}^d)\) with \(1\le p\le \infty \) by constructing a sequence of initial data for showing discontinuity of the solution map at zero. As a by-product, we demonstrate that the Cauchy problem for the incompressible Navier–Stokes equations is also ill-posed in \(\dot{B}_{p,\infty }^{d/p-1}(\mathbb {R}^d)\), which is an interesting open problem in itself.